Musical Scales with Pythagorean Intervals

Abstract

The article examines the arithmetical qualities of musical scales tuned unequally with
elementary intervals of the breadth of 0.98045 of a semitone (a “diminished” semitone)
and 1.01955 of a semitone (an “augmented” semitone). It is demonstrated that each interval
comprising a sum or a variety of a whole number of perfect fifths and the whole number of
perfect octaves (in other words, a Pythagorean interval) may be presented as a sum of diminished
and augmented semitones. Thereby, the Pythagorean major second is equal in sum to three
augmented semitones, a minor Pythagorean third is equal in sum to two diminished semitones, a
major Pythagorean third is equal in sum to four augmented semitones, a perfect fourth is equal inand augmented semitones as elementary intervals of the scale the opportunity arises to build
musical scales containing other Pythagorean intervals, besides the octave.

Keywords: musical intervals, musical scales, tuning the piano.
sum to three diminished and two augmented semitones, etc. Thereby, with the use of diminished